To complete the square for the function \( f(x) = x^2 - 12x - 29 \), follow these steps:
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Take the linear term (-12) and halve it: \[ -\frac{12}{2} = -6 \]
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Square this result: \[ (-6)^2 = 36 \]
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Add and subtract this square inside the function: \[ f(x) = (x^2 - 12x + 36) - 36 - 29 \]
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Rewrite the quadratic part as a perfect square: \[ f(x) = (x - 6)^2 - 36 - 29 \]
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Combine the constants: \[ -36 - 29 = -65 \]
Putting it all together, we have: \[ f(x) = (x - 6)^2 - 65 \]
Thus, the completed square form is: \[ f(x) = (x - 6)^2 - 65 \]
So, you can fill the blanks as follows: \[ f(x) = (x - 6)^2 - 65 \] Thus: \[ f(x) = (x + \mathbf{(-6)})^2 + \mathbf{(-65)} \]